Find the points on the curve y = x3 – 2x2 – x at which the tangent lines are parallel to the line y = 3x – 2.

Asked by Topperlearning User | 7th Aug, 2014, 09:58: AM

Expert Answer:

L e t space P left parenthesis x subscript 1 comma y subscript 1 right parenthesis space b e space t h e space r e q u i r e d space p o i n t. space T h e space g i v e n space c u r v e space i s
y equals x cubed minus 2 x squared minus x
rightwards double arrow fraction numerator d y over denominator d x end fraction equals 3 x squared minus 4 x minus 1
rightwards double arrow open parentheses fraction numerator d y over denominator d x end fraction close parentheses subscript left parenthesis x subscript 1 comma y subscript 1 right parenthesis end subscript equals 3 x subscript 1 squared minus 4 x subscript 1 minus 1
S i n c e space t h e space tan g e n t space a t space left parenthesis x subscript 1 comma y subscript 1 right parenthesis space i s space p a r a l l e l space t o space t h e space l i n e space y equals 3 x minus 2
therefore S l o p e space o f space t h e space tan g e n t space a t space left parenthesis x subscript 1 comma y subscript 1 right parenthesis equals s l o p e space o f space t h e space l i n e comma space y equals 3 x minus 2
therefore open parentheses fraction numerator d y over denominator d x end fraction close parentheses subscript left parenthesis x subscript 1 comma y subscript 1 right parenthesis end subscript equals 3
rightwards double arrow 3 x subscript 1 squared minus 4 x subscript 1 minus 1 equals 3
rightwards double arrow 3 x subscript 1 squared minus 4 x subscript 1 minus 4 equals 0
rightwards double arrow left parenthesis 3 x subscript 1 plus 2 right parenthesis left parenthesis x subscript 1 minus 2 right parenthesis equals 0
therefore x subscript 1 equals 2 comma space minus 2 over 3
S i n c e space left parenthesis x subscript 1 comma y subscript 1 right parenthesis space l i e s space o n space t h e space c u r v e left parenthesis i right parenthesis. space T h e r e f o r e
y subscript 1 equals x subscript 1 cubed minus 2 x subscript 1 squared minus x subscript 1
N o w comma space x subscript 1 equals 2 rightwards double arrow y subscript 1 equals 2 cubed minus 2 cross times 4 minus 2 equals minus 2
space x subscript 1 equals minus 2 over 3 rightwards double arrow y subscript 1 equals open parentheses minus 2 over 3 close parentheses cubed minus 2 cross times open parentheses minus 2 over 3 close parentheses squared plus 2 over 3 equals minus 14 over 27
T h e space r e q u i r e d space p o i n t s space a r e space left parenthesis 2 comma minus 2 right parenthesis space a n d space open parentheses minus 2 over 3 comma minus 14 over 27 close parentheses

Answered by  | 7th Aug, 2014, 11:58: AM